| Summary: | Using the direct variational method together with the monotonicity approach we consider the existence of non-spurious solutions to the following Dirichlet problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mfenced open="(" close=")"><mi>t</mi></mfenced><mo> </mo><mo>=</mo><mi>f</mi><mfenced separators="" open="(" close=")"><mi>t</mi><mo>,</mo><mi>x</mi><mfenced open="(" close=")"><mi>t</mi></mfenced></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mfenced open="(" close=")"><mn>0</mn></mfenced><mo> </mo><mo>=</mo><mi>x</mi><mfenced open="(" close=")"><mn>1</mn></mfenced><mo> </mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mo> </mo><mfenced separators="" open="[" close="]"><mn>0</mn><mo>,</mo><mn>1</mn></mfenced><mo> </mo><mo>×</mo><mo> </mo><mi mathvariant="double-struck">R</mi><mo>→</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> is a jointly continuous and not necessarily convex function. A new approach towards deriving the discrete family of approximating problems is proposed.
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